Clustering analysis

The aim of this study was to determine whether there are any clusters inside the venus or not. As the clustering algorithm K-means++ was chosen due to its simplicity and efficiency. As the main metric for choosing the correct number of clusters (K), WSS (within clusters sums of squares) and silhouette score were chosen. (tady muzeme citovat napr: https://www.oreilly.com/library/view/practical-statistics-for/9781491952955/ (strana 294-304) a asi by chtelo ocitovat i tu knihovnu pro ten algorithmus (https://scikit-learn.org/stable/modules/generated/sklearn.cluster.KMeans.html)). At first, clustering was applied to all particles no matter what size. This approach did not bring any significant results as the size of the most particles is extremely small and on top of that, there was a huge overlap among individual clusters. Afterwards, we have decided to take a more reasonable approach by focusing solely on particles with volume greater than or equal to 2 mm^3 (in total 69 particles). This threshold was set as the minimum possible particle volume (size) that could have been exposed to the intentional human manipulation in terms of technological or symbolical purposes. Firstly, only particle coordinates (BaryCenters) were fed into the algorithm. Based on the elbow method (using WSS metric) 2 clusters were designated. None of them brought any statistically significant result. We failed to reject all our null hypothesis related to physical features. As the main statistical test, two sample t-test was chosen to compare mean values of two distinct clusters, where first one contains 37 particles and the second on 32. The significant level (alpha) was set to 0.05 (5%). Anisotropy mean values were equal to 0.769 and 0.727 with p-value = 0.212, elongation was 0.518 and 0.526 with p-value = 0.872, flatness was 0.463 and 0.523 with p-value = 0.267 and sphericity has means equal to 0.386 and 0.415 with p-value = 0.144. Subsequently, we tried to group our particles based on their physical features. At first a PCA analysis was applied to standardised (z-score values) features from particles with volume 2 mm^3 and more. Only two components were taken into consideration as they contain together more than 80% of the total variance. Using the elbow method 3 clusters were singled out by applying K-means++ algortihm to before gained PCA scores. Given that cluster labels were attached to our original data and plot using 3D scatterplot. While the first and second clusters overlaped, the third cluster created a fairly homogeneous formation mostly in the area of the upper torso of venus. This particular cluster was distincted by high avarage value of anisotropy (0.880) and very low avarage value of flatness (0.251). Since our particle volume threshold contained only 69 particles in total, a decition was made to move our lower limit volume boundary from 2 mm^3 to 1 mm^3 in order to be able to include more particles into our analysis in terms of reducing the uncertainty of our result. The same procedure as for the particles greater than or equal to 2 mm^3 was carried out on particles including all data records with volume higher than 1 mm^3. The result clusters were far less distinct from each other from a physical perspective and their spatial distribution was rather evenly spaced without any distinct agglomerations across the figurine's body. Considering that k-means algorithm belong to so called non-deterministic (stochastic) methods and the result interpretability and the performance evaluation is not as trivial as in the case of supervised learning algorithms . As the third approach to tackle our clustering conandrum was to divide the figurine into 6 equal sized parts along vertical axes (using BarycenterZ coordinates). Particular attention was given again to particles with volume greater than or equal to 2 mm^3. Mean values of physical features were very similar across all slices and the ratio of particles with the volume greater than or equal to 2 mm^3 to all particles in a particular part of the venus were ranging between 0.34 % (the head part slice) to 0.82% (the upper torso part) without any steep fluctuation. Bearing in mind all previous methods we can claim that is unlikely that in the process of making the figuring were any internal structures created intentionally with any clear purpose.

import pandas as pd
import seaborn as sns 
import plotly.express as px
import matplotlib.pyplot as plt
import numpy as np

#clustering 
from sklearn.preprocessing import StandardScaler
from sklearn.cluster import KMeans
from scipy.stats import stats 
from sklearn.metrics import silhouette_score
from sklearn.decomposition import PCA

df = pd.read_csv('C:/Users/hamro/Desktop/final_table.csv')

all_particles = df.copy()

all particles

all_particles

	Volume3d (mm^3)	BaryCenterX (mm)	BaryCenterY (mm)	BaryCenterZ (mm)	Anisotropy	Elongation	Flatness	VoxelFaceArea (mm^2)	Sphericity	SurfaceParticles
0	91.752900	-1249.26	-1243.58	-1685.28	0.545994	0.731435	0.620706	244.736000	0.401975	False
1	33.285200	-1241.94	-1257.46	-1686.12	0.946436	0.566783	0.094506	175.078000	0.285816	False
2	26.857600	-1246.98	-1235.86	-1648.17	0.862477	0.343091	0.400835	194.946000	0.222475	False
3	20.575500	-1249.24	-1250.28	-1680.69	0.844642	0.736602	0.210912	94.278900	0.385156	False
4	17.937600	-1242.87	-1249.47	-1647.25	0.710594	0.471159	0.614242	71.309800	0.464705	False
...	...	...	...	...	...	...	...	...	...	...
102510	0.000007	-1245.75	-1250.92	-1618.70	-1.000000	-1.000000	-1.000000	0.002151	0.805997	False
102511	0.000007	-1245.36	-1250.98	-1618.53	1.000000	0.000000	-1.000000	0.002151	0.805997	False
102512	0.000007	-1247.38	-1249.95	-1618.51	-1.000000	-1.000000	-1.000000	0.002151	0.805997	False
102513	0.000007	-1244.28	-1245.09	-1618.51	-1.000000	-1.000000	-1.000000	0.002151	0.805997	True
102514	0.000007	-1244.05	-1248.17	-1618.21	-1.000000	-1.000000	-1.000000	0.002151	0.805997	False

102515 rows × 10 columns

all particles - descriptive stats

all_particles.describe()

	Volume3d (mm^3)	BaryCenterX (mm)	BaryCenterY (mm)	BaryCenterZ (mm)	Anisotropy	Elongation	Flatness	VoxelFaceArea (mm^2)	Sphericity
count	102515.000000	102515.000000	102515.000000	102515.000000	102515.000000	102515.000000	102515.000000	102515.000000	102515.000000
mean	0.009713	-1246.522981	-1246.863327	-1672.536396	0.676258	0.438560	0.328198	0.146898	0.658045
std	0.361543	4.688294	7.830213	25.611063	0.416351	0.406608	0.508212	1.638454	0.080128
min	0.000007	-1262.260000	-1269.460000	-1731.770000	-1.000000	-1.000000	-1.000000	0.002151	0.167262
25%	0.000048	-1249.780000	-1252.495000	-1691.630000	0.611283	0.292913	0.179971	0.009320	0.614133
50%	0.000170	-1246.460000	-1246.840000	-1672.400000	0.772373	0.499862	0.447890	0.022224	0.659861
75%	0.000618	-1243.100000	-1241.280000	-1654.175000	0.904540	0.692323	0.655831	0.055918	0.707018
max	91.752900	-1233.450000	-1225.980000	-1617.670000	1.000000	0.999980	1.000000	244.736000	0.805997

1: clutering only particles with volume greater than or equal to 2mm^3

particles_over_2 = all_particles[all_particles['Volume3d (mm^3) '] > 2]

particles_over_2.describe()

	Volume3d (mm^3)	BaryCenterX (mm)	BaryCenterY (mm)	BaryCenterZ (mm)	Anisotropy	Elongation	Flatness	VoxelFaceArea (mm^2)	Sphericity
count	69.000000	69.000000	69.000000	69.000000	69.000000	69.000000	69.000000	69.000000	69.000000
mean	7.194427	-1246.532464	-1247.985072	-1673.264203	0.749591	0.522337	0.490613	43.186864	0.399711
std	11.879673	5.010710	8.228589	25.167699	0.138979	0.190592	0.223466	40.252915	0.082241
min	2.057330	-1260.040000	-1263.090000	-1720.930000	0.373645	0.141648	0.094506	15.814000	0.222475
25%	2.649340	-1249.400000	-1254.750000	-1692.350000	0.659225	0.399644	0.304423	22.947400	0.334487
50%	3.763380	-1246.280000	-1247.770000	-1669.500000	0.762504	0.492984	0.491434	29.387900	0.406799
75%	7.212370	-1242.330000	-1242.440000	-1655.430000	0.860182	0.672624	0.626743	44.775600	0.464705
max	91.752900	-1236.340000	-1229.200000	-1619.440000	0.966510	0.917335	0.959761	244.736000	0.557364

particles_over_2.columns

Index(['Volume3d (mm^3) ', 'BaryCenterX (mm) ', 'BaryCenterY (mm) ',
       'BaryCenterZ (mm) ', 'Anisotropy', 'Elongation', 'Flatness',
       'VoxelFaceArea (mm^2)', 'Sphericity', 'SurfaceParticles'],
      dtype='object')

2: 1. particles greater than or equal to 2 mm^3; clustering based on coordinates

# number or possible clusters is equal to the number of points
x_coord = particles_over_2[['BaryCenterX (mm) ', 'BaryCenterY (mm) ',
       'BaryCenterZ (mm) ']]
wcss_clust = []
for i in range(1,10):
    kmeans = KMeans(i, init="k-means++", random_state=42)
    kmeans.fit(x_coord)
    wcss_clust.append(kmeans.inertia_)

plt.plot(range(1,10), wcss_clust, marker = '*')

[<matplotlib.lines.Line2D at 0x2bb6d8ec370>]

#silhuette score
for i in range(2,10):
    k_means_three = KMeans(n_clusters = i)
    k_means_three.fit(x_coord)
    print(f'WSS for K={i}: {k_means_three.inertia_}')
    labels_three = k_means_three.labels_
    #print(labels_three)
    print(f"{i}: {silhouette_score(x_coord, labels_three)}")

WSS for K=2: 18325.24549746622
2: 0.5036219719610647
WSS for K=3: 12557.427502093411
3: 0.3773326898682717
WSS for K=4: 9426.49948399604
4: 0.3440940235018148
WSS for K=5: 7461.9943561660475
5: 0.3474167172506071
WSS for K=6: 6570.795314660886
6: 0.3228409740059239
WSS for K=7: 5598.258385541129
7: 0.3106010471705073
WSS for K=8: 4786.76374264069
8: 0.30915176843079384
WSS for K=9: 4255.457309285719
9: 0.3372788252899918

based on elbow method and silhouette score two clusters were identified as the ideal K

k_means_coordinates_2 = KMeans(n_clusters=2, init="k-means++", random_state=42)

k_means_coordinates_2.fit(x_coord)

KMeans(n_clusters=2, random_state=42)

particles_over_2['two_clust_coord'] = k_means_coordinates_2.labels_

particles_over_2.head()

	Volume3d (mm^3)	BaryCenterX (mm)	BaryCenterY (mm)	BaryCenterZ (mm)	Anisotropy	Elongation	Flatness	VoxelFaceArea (mm^2)	Sphericity	SurfaceParticles	two_clust_coord
0	91.7529	-1249.26	-1243.58	-1685.28	0.545994	0.731435	0.620706	244.7360	0.401975	False	1
1	33.2852	-1241.94	-1257.46	-1686.12	0.946436	0.566783	0.094506	175.0780	0.285816	False	1
2	26.8576	-1246.98	-1235.86	-1648.17	0.862477	0.343091	0.400835	194.9460	0.222475	False	0
3	20.5755	-1249.24	-1250.28	-1680.69	0.844642	0.736602	0.210912	94.2789	0.385156	False	1
4	17.9376	-1242.87	-1249.47	-1647.25	0.710594	0.471159	0.614242	71.3098	0.464705	False	0

particles_over_2.columns

Index(['Volume3d (mm^3) ', 'BaryCenterX (mm) ', 'BaryCenterY (mm) ',
       'BaryCenterZ (mm) ', 'Anisotropy', 'Elongation', 'Flatness',
       'VoxelFaceArea (mm^2)', 'Sphericity', 'SurfaceParticles',
       'two_clust_coord'],
      dtype='object')

import plotly.express as px

fig = px.scatter_3d(particles_over_2, x='BaryCenterX (mm) ', y='BaryCenterY (mm) ', z='BaryCenterZ (mm) ',
              color='two_clust_coord')
fig.show()

group_2_over_2 = particles_over_2.groupby(['two_clust_coord']).mean()

# number of clusters
group_2_over_2['CountTwoSegment'] = particles_over_2[['two_clust_coord','Sphericity']].groupby(['two_clust_coord']).count()

group_2_over_2.rename({0:'top-cluster',1:'bottom-cluster'})

	Volume3d (mm^3)	BaryCenterX (mm)	BaryCenterY (mm)	BaryCenterZ (mm)	Anisotropy	Elongation	Flatness	VoxelFaceArea (mm^2)	Sphericity	SurfaceParticles	CountTwoSegment
two_clust_coord
top-cluster	6.217747	-1247.474324	-1248.954324	-1653.580000	0.769289	0.518836	0.462868	42.818478	0.386143	0.0	37
bottom-cluster	8.323714	-1245.443438	-1246.864375	-1696.024062	0.726815	0.526384	0.522692	43.612809	0.415398	0.0	32

fig = plt.figure(figsize=(12,8))

plt.subplot(2, 2, 1)
sns.boxplot(data=particles_over_2, x='two_clust_coord',  y='Anisotropy')
plt.title("Anisotropy")
plt.xlabel('clusters', {'size':12})
plt.ylabel('Anisotropy', {'size':12})

plt.subplot(2, 2, 2)
sns.boxplot(data=particles_over_2, x='two_clust_coord',  y='Elongation')
plt.title("Elongation")
plt.xlabel('clusters', {'size':12})
plt.ylabel('Elongation', {'size':12})

plt.subplot(2, 2, 3)
sns.boxplot(data=particles_over_2, x='two_clust_coord',  y='Flatness')
plt.title("Flatness")
plt.xlabel('clusters', {'size':12})
plt.ylabel('Flatness', {'size':12})

plt.subplot(2, 2, 4)
sns.boxplot(data=particles_over_2, x='two_clust_coord',  y='Sphericity')
plt.title("Sphericity")
plt.xlabel('clusters', {'size':12})
plt.ylabel('Sphericity', {'size':12})
plt.tight_layout(pad=2)
plt.show()

2 sample t-test, respectively we are going to compare cluster means for anisotropy, elongation, flatness and sphericity

significance level is set to 0.05

H0 - there is no difference between means of two clusters

clust_0 = particles_over_2[particles_over_2['two_clust_coord'] == 0]
clust_1 = particles_over_2[particles_over_2['two_clust_coord'] == 1]

clust_0_anis = clust_0['Anisotropy']
clust_1_anis = clust_1['Anisotropy']
res_anis = stats.ttest_ind(a= clust_0_anis,
                b= clust_1_anis,
                equal_var=False)
print("for anisotropy we cannot reject null hypothesis, p val is: ",res_anis[1])

for anisotropy we cannot reject null hypothesis, p val is:  0.21267331301420572

clust_0_elong = clust_0['Elongation']
clust_1_elong = clust_1['Elongation']
res_elong = stats.ttest_ind(a= clust_0_elong,
                b= clust_1_elong,
                equal_var=False)
print("for elongation we cannot reject null hypothesis, p val is: ",res_elong[1])

for elongation we cannot reject null hypothesis, p val is:  0.8722666583663269

clust_0_flat = clust_0['Flatness']
clust_1_flat = clust_1['Flatness']
res_flat = stats.ttest_ind(a= clust_0_flat,
                b= clust_1_flat,
                equal_var=False)
print("for flatness we cannot reject null hypothesis, p val is: ",res_flat[1])

for flatness we cannot reject null hypothesis, p val is:  0.2672472735453102

clust_0_spher = clust_0['Sphericity']
clust_1_spher = clust_1['Sphericity']
res_spher = stats.ttest_ind(a= clust_0_spher,
                b= clust_1_spher,
                equal_var=False)
print("for sphericity we cannot reject null hypothesis, p val is: ",res_spher[1])

for sphericity we cannot reject null hypothesis, p val is:  0.1447081316172907

2. 2., particles greater than or equal to 2 mm^3; clustering based on physical features (anisotropy, elongation, flatness, sphericity)

PCA + K-means++

particles_over_2 = all_particles[all_particles['Volume3d (mm^3) '] > 2]

particles_over_2.columns

Index(['Volume3d (mm^3) ', 'BaryCenterX (mm) ', 'BaryCenterY (mm) ',
       'BaryCenterZ (mm) ', 'Anisotropy', 'Elongation', 'Flatness',
       'VoxelFaceArea (mm^2)', 'Sphericity', 'SurfaceParticles'],
      dtype='object')

physical_features = particles_over_2[['Anisotropy', 'Elongation', 'Flatness','Sphericity']]

# standardize - z scaling

scaler = StandardScaler()
physical_features_scaled = scaler.fit_transform(physical_features)

physical_features_scaled = pd.DataFrame(physical_features_scaled, columns=['Anisotropy', 'Elongation', 'Flatness','Sphericity'])

physical_features_scaled.head()

	Anisotropy	Elongation	Flatness	Sphericity
0	-1.475684	1.105135	0.586428	0.027741
1	1.426747	0.234909	-1.785549	-1.395032
2	0.818206	-0.947357	-0.404694	-2.170863
3	0.688937	1.132444	-1.260819	-0.178272
4	-0.282652	-0.270487	0.557290	0.796088

#initialize pca
pca = PCA()

pca.fit(physical_features_scaled)

PCA()

pca.explained_variance_ratio_

array([0.52233048, 0.29288467, 0.17665369, 0.00813115])

two components were chosen as they capture more than 80% of all variability

plt.title("explained variance by components")
plt.plot(range(1,5), pca.explained_variance_ratio_.cumsum(), marker='o', linestyle='--')
plt.show()

pca = PCA(n_components=2)

pca.fit_transform(physical_features_scaled)

array([[ 1.69784283, -0.21790518],
       [-2.28213024, -1.18545698],
       [-2.05896506,  0.80972856],
       [-0.66083809, -1.65873864],
       [ 0.68468306,  0.40273266],
       [ 0.67331076, -0.89001326],
       [-2.0922322 , -0.8317282 ],
       [-0.83328295, -0.77997422],
       [-1.02503825, -0.73004263],
       [-2.17649658,  1.78442038],
       [ 1.08190551,  0.11940239],
       [ 0.37359159,  0.09949121],
       [-1.72175425, -0.49947993],
       [ 0.95149502,  1.20933389],
       [ 1.36880417, -1.787673  ],
       [ 0.15276168, -0.30836946],
       [-0.11358878,  0.68547152],
       [ 0.71432872,  0.46914567],
       [ 1.26781765, -0.89571814],
       [-1.2759528 , -1.64637412],
       [-1.02697378,  0.0889843 ],
       [ 2.09248455, -1.2877939 ],
       [-0.02721169,  1.36540731],
       [-1.50421012,  0.04033337],
       [-0.02996312, -2.19697457],
       [ 0.40560247,  0.05797175],
       [-0.72345217, -0.09227255],
       [-0.79683288, -0.87805725],
       [-0.56286975, -0.12161921],
       [-0.38188137,  0.58379119],
       [ 0.65409647,  1.76831451],
       [ 0.8661913 ,  1.74426478],
       [ 0.98858131, -0.84308625],
       [-1.06713043, -1.72632957],
       [-2.11327511, -0.85978317],
       [ 0.86393053, -1.05753389],
       [-2.48748842,  1.10995769],
       [ 0.02226271,  0.85051381],
       [ 1.58523841,  1.92671667],
       [ 0.19180312,  1.21589782],
       [-1.52090881,  0.05371981],
       [-0.75117976, -1.01530257],
       [ 1.82999147, -0.75843357],
       [ 3.28132679, -1.11136432],
       [ 1.04988978,  0.16371396],
       [ 3.60852521,  0.48248536],
       [ 1.62949117, -1.1331376 ],
       [ 1.40920572, -1.71189537],
       [ 0.06084942,  0.07423134],
       [-2.13382819, -0.07963707],
       [ 1.12763133,  1.29159188],
       [-0.33797579,  0.93530044],
       [ 0.11657326,  0.68267733],
       [-1.32825871, -0.90067298],
       [-1.52216622,  2.31382616],
       [ 2.85482703,  1.04226722],
       [ 0.02891181,  1.95226262],
       [ 2.47547768,  0.12693041],
       [-1.54458355, -1.10749824],
       [ 1.22989886,  0.42445743],
       [-0.04460979,  1.55920017],
       [-1.03968957,  1.97058805],
       [-0.89686485, -1.00590699],
       [-1.4853406 , -0.1238751 ],
       [ 1.20780799, -0.82698988],
       [ 0.17335248,  0.06098017],
       [ 1.76152985, -0.37896035],
       [ 0.14633077,  1.1899492 ],
       [-3.06137854, -0.00746286]])

loadings = pd.DataFrame(data=pca.components_,
                       columns=physical_features_scaled.columns,
                       index=['component1','component2'])

loadings - correlation between pca components and original features

loadings

	Anisotropy	Elongation	Flatness	Sphericity
component1	-0.662029	0.398279	0.457936	0.439757
component2	-0.095712	-0.682150	0.684400	-0.238971

fig = plt.figure(figsize=(12,8))
sns.heatmap(loadings, cmap='RdBu', annot=True, vmin=-1, vmax=1)
plt.title("correlation between components and original physical features", size=14)
plt.yticks(rotation=45, size=12)
plt.xticks(size=12)
plt.tight_layout()
plt.show()

scores_pca = pca.fit_transform(physical_features_scaled)

pca scores

print(scores_pca.shape)
scores_pca[0:5]

(69, 2)

array([[ 1.69784283, -0.21790518],
       [-2.28213024, -1.18545698],
       [-2.05896506,  0.80972856],
       [-0.66083809, -1.65873864],
       [ 0.68468306,  0.40273266]])

wcss_clust_pca = []
for i in range(1,10):
    kmeans = KMeans(i, init="k-means++", random_state=42)
    kmeans.fit(scores_pca)
    wcss_clust_pca.append(kmeans.inertia_)

based on the elbow method we're choosing 3 clusters

plt.plot(range(1,10), wcss_clust_pca, marker = '*')
plt.title("pca with k means clustering")
plt.show()

#silhuette score
for i in range(2,10):
    k_means_three = KMeans(n_clusters = i)
    k_means_three.fit(scores_pca)
    print(f'WSS for K={i}: {k_means_three.inertia_}')
    labels_three = k_means_three.labels_
    #print(labels_three)
    print(f"{i}: {silhouette_score(scores_pca, labels_three)}")

WSS for K=2: 124.83301256220528
2: 0.39990144134364825
WSS for K=3: 78.54270432033
3: 0.42321903979946557
WSS for K=4: 59.423630989337006
4: 0.4398274611987941
WSS for K=5: 48.89965185973766
5: 0.43945526448835415
WSS for K=6: 39.33140946590044
6: 0.39609904895579723
WSS for K=7: 30.453313410405784
7: 0.40014479991212426
WSS for K=8: 26.626543332921912
8: 0.3992686126593644
WSS for K=9: 22.492480945762843
9: 0.41380671227178123

k_means_physical = KMeans(n_clusters=3, init="k-means++", random_state=42)

k_means_physical.fit(scores_pca)

KMeans(n_clusters=3, random_state=42)

k_means_physical.labels_

array([0, 2, 2, 2, 1, 0, 2, 2, 2, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 2, 2, 0,
       1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 0, 2, 2, 0, 2, 1, 1, 1, 2, 2, 0, 0,
       0, 0, 0, 0, 1, 2, 1, 1, 1, 2, 1, 0, 1, 0, 2, 0, 1, 1, 2, 2, 0, 1,
       0, 1, 2])

K-means++ clustering with PCA results

particles_over_2.columns

Index(['Volume3d (mm^3) ', 'BaryCenterX (mm) ', 'BaryCenterY (mm) ',
       'BaryCenterZ (mm) ', 'Anisotropy', 'Elongation', 'Flatness',
       'VoxelFaceArea (mm^2)', 'Sphericity', 'SurfaceParticles'],
      dtype='object')

part_over_2 = particles_over_2[['Volume3d (mm^3) ', 'BaryCenterX (mm) ', 'BaryCenterY (mm) ',
       'BaryCenterZ (mm) ','Anisotropy', 'Elongation', 'Flatness', 'Sphericity', 
       ]]

pca.score

<bound method PCA.score of PCA(n_components=2)>

df_pca_scores = pd.DataFrame(scores_pca, columns=['Component 1', 'Component 2'])
k_means_pca_clusters = pd.DataFrame(k_means_physical.labels_, columns=["PcaClusters"])

df_segm_pca = pd.concat([part_over_2.reset_index(drop=True), df_pca_scores , k_means_pca_clusters], axis=1)

table of particles with the volume greater than or equal to 2 mm^3, 2 pca component created based on physical features (anisotropy, elongation, flatness, sphericity) and based on the 3 clusters were designated by using K means++

df_segm_pca.head()

	Volume3d (mm^3)	BaryCenterX (mm)	BaryCenterY (mm)	BaryCenterZ (mm)	Anisotropy	Elongation	Flatness	Sphericity	Component 1	Component 2	PcaClusters
0	91.7529	-1249.26	-1243.58	-1685.28	0.545994	0.731435	0.620706	0.401975	1.697843	-0.217905	0
1	33.2852	-1241.94	-1257.46	-1686.12	0.946436	0.566783	0.094506	0.285816	-2.282130	-1.185457	2
2	26.8576	-1246.98	-1235.86	-1648.17	0.862477	0.343091	0.400835	0.222475	-2.058965	0.809729	2
3	20.5755	-1249.24	-1250.28	-1680.69	0.844642	0.736602	0.210912	0.385156	-0.660838	-1.658739	2
4	17.9376	-1242.87	-1249.47	-1647.25	0.710594	0.471159	0.614242	0.464705	0.684683	0.402733	1

mean values for different clusters

df_mean_segm_pca = df_segm_pca.groupby('PcaClusters').mean()

df_mean_segm_pca

	Volume3d (mm^3)	BaryCenterX (mm)	BaryCenterY (mm)	BaryCenterZ (mm)	Anisotropy	Elongation	Flatness	Sphericity	Component 1	Component 2
PcaClusters
0	9.181104	-1245.360526	-1243.791053	-1674.727895	0.590278	0.726338	0.573164	0.462748	1.703824	-0.554803
1	4.921554	-1246.346400	-1248.516000	-1680.404400	0.740282	0.392625	0.667039	0.393894	0.104483	1.035428
2	7.957426	-1247.609200	-1250.641600	-1665.011600	0.879977	0.497008	0.251447	0.357618	-1.399389	-0.613779

fig = plt.figure(figsize=(12,8))

plt.subplot(2, 2, 1)
sns.boxplot(data=df_segm_pca, x='PcaClusters',  y='Anisotropy')
plt.title("Anisotropy")
plt.xlabel('clusters', {'size':12})
plt.ylabel('Anisotropy', {'size':12})

plt.subplot(2, 2, 2)
sns.boxplot(data=df_segm_pca, x='PcaClusters',  y='Elongation')
plt.title("Elongation")
plt.xlabel('clusters', {'size':12})
plt.ylabel('Elongation', {'size':12})

plt.subplot(2, 2, 3)
sns.boxplot(data=df_segm_pca, x='PcaClusters',  y='Flatness')
plt.title("Flatness")
plt.xlabel('clusters', {'size':12})
plt.ylabel('Flatness', {'size':12})

plt.subplot(2, 2, 4)
sns.boxplot(data=df_segm_pca, x='PcaClusters',  y='Sphericity')
plt.title("Sphericity")
plt.xlabel('clusters', {'size':12})
plt.ylabel('Sphericity', {'size':12})
plt.tight_layout(pad=2)
plt.show()

# number of clustersdf_segm_pca.groupby('PcaClusters')
df_mean_segm_pca['CountThreeSegment'] = df_segm_pca[['PcaClusters','Sphericity']].groupby(['PcaClusters']).count()

three clusters: 0: 19 particles, 1: 25 particles, 2: 25 particles

df_mean_segm_pca

	Volume3d (mm^3)	BaryCenterX (mm)	BaryCenterY (mm)	BaryCenterZ (mm)	Anisotropy	Elongation	Flatness	Sphericity	Component 1	Component 2	CountThreeSegment
PcaClusters
0	9.181104	-1245.360526	-1243.791053	-1674.727895	0.590278	0.726338	0.573164	0.462748	1.703824	-0.554803	19
1	4.921554	-1246.346400	-1248.516000	-1680.404400	0.740282	0.392625	0.667039	0.393894	0.104483	1.035428	25
2	7.957426	-1247.609200	-1250.641600	-1665.011600	0.879977	0.497008	0.251447	0.357618	-1.399389	-0.613779	25

x_axis = df_segm_pca['Component 1']
y_axis = df_segm_pca['Component 2']
fig = plt.figure(figsize=(12,8))
sns.scatterplot(x_axis,y_axis,hue=df_segm_pca['PcaClusters'], palette=['g','r','b'])
plt.title("clusters by PCA components", size=14)
plt.show()

C:\Users\hamro\Anaconda3\lib\site-packages\seaborn\_decorators.py:36: FutureWarning:

Pass the following variables as keyword args: x, y. From version 0.12, the only valid positional argument will be `data`, and passing other arguments without an explicit keyword will result in an error or misinterpretation.

df_segm_pca

	Volume3d (mm^3)	BaryCenterX (mm)	BaryCenterY (mm)	BaryCenterZ (mm)	Anisotropy	Elongation	Flatness	Sphericity	Component 1	Component 2	PcaClusters
0	91.75290	-1249.26	-1243.58	-1685.28	0.545994	0.731435	0.620706	0.401975	1.697843	-0.217905	0
1	33.28520	-1241.94	-1257.46	-1686.12	0.946436	0.566783	0.094506	0.285816	-2.282130	-1.185457	2
2	26.85760	-1246.98	-1235.86	-1648.17	0.862477	0.343091	0.400835	0.222475	-2.058965	0.809729	2
3	20.57550	-1249.24	-1250.28	-1680.69	0.844642	0.736602	0.210912	0.385156	-0.660838	-1.658739	2
4	17.93760	-1242.87	-1249.47	-1647.25	0.710594	0.471159	0.614242	0.464705	0.684683	0.402733	1
...	...	...	...	...	...	...	...	...	...	...	...
64	2.15399	-1237.99	-1243.50	-1703.01	0.629185	0.754557	0.491434	0.425615	1.207808	-0.826990	0
65	2.10846	-1250.47	-1256.48	-1666.34	0.687320	0.597981	0.522894	0.334487	0.173352	0.060980	1
66	2.09733	-1260.04	-1242.14	-1665.96	0.603934	0.663882	0.596591	0.501057	1.761530	-0.378960	0
67	2.07964	-1243.71	-1244.93	-1637.44	0.741252	0.371061	0.697319	0.399350	0.146331	1.189949	1
68	2.05733	-1247.83	-1246.52	-1676.76	0.966510	0.277820	0.120545	0.261976	-3.061379	-0.007463	2

69 rows × 11 columns

import plotly.express as px

fig = px.scatter_3d(df_segm_pca, x='BaryCenterX (mm) ', y='BaryCenterY (mm) ', z='BaryCenterZ (mm) ',
              color='PcaClusters', symbol='PcaClusters', opacity=0.7)
fig.update_layout(margin=dict(l=0, r=0, b=0, t=0))
fig.show()

as we can see above there is slightly distinguished yellow cluster number 2, typical of high value of anisotropy, mean (0,89)

2.3. clustering - particles greater than or equal to 1 mm^3

trying to gain more insight into these clusters we can try to include all particles with volume greater than or equal to 2 mm^3

all particles greater than or equal to 1 mm^3 clustering using PCA on their physical attributes

particles_over_1 = all_particles[all_particles['Volume3d (mm^3) '] > 1]

physical_features_1 = particles_over_1[['Anisotropy', 'Elongation', 'Flatness','Sphericity']]
scaler = StandardScaler()
physical_features_scaled_ = scaler.fit_transform(physical_features_1)

physical_features_scaled_

array([[-1.5660082 ,  1.13293305,  0.59333438, -0.21215505],
       [ 1.58250012,  0.22499242, -1.956182  , -1.60010257],
       [ 0.92236554, -1.00851252, -0.47197234, -2.35694302],
       [ 0.78213638,  1.16142544, -1.39217668, -0.41312541],
       [-0.2718271 , -0.30230706,  0.56201536,  0.53738517],
       [-0.48502103,  1.1154086 , -0.30824038, -0.15451719],
       [ 1.33624403,  0.0417081 , -1.64324236, -1.65278695],
       [ 0.66013268,  0.3023746 , -0.98864399, -0.61476226],
       [ 1.05899364, -0.31181372, -1.17402406,  0.19515138],
       [ 1.38669822, -2.10923861,  0.23570459, -1.72577182],
       [-1.32920294,  0.88067585,  0.58115368, -1.0327862 ],
       [-0.13940548, -0.09983849,  0.18618721,  0.21030422],
       [ 1.23657793, -0.47168984, -1.34084722, -0.81692215],
       [-0.67572011, -0.4914586 ,  1.36543234, -0.01667645],
       [-0.72361896,  2.04966348, -0.54189262,  0.49498376],
       [ 0.23100865, -0.11585753, -0.25087872,  0.64854361],
       [ 0.12305539, -0.63567962,  0.4076103 , -0.27060883],
       [-0.43834087, -0.18195745,  0.63840875,  0.203607  ],
       [-0.86648209,  1.24425575, -0.06095391,  0.305042  ],
       [ 0.94293406,  1.38109312, -1.57222708, -1.5758077 ],
       [ 0.66484237, -0.49965285, -0.51911075, -0.80264928],
       [-1.2848108 ,  1.68018328,  0.02543017,  1.11728956],
       [-0.56797914, -0.37417494,  1.04504233, -1.83841531],
       [ 0.94766733, -0.6542573 , -0.81654447, -1.06639313],
       [ 0.70023972,  1.60196292, -1.43035641,  0.53851754],
       [-0.31627428,  0.12728453,  0.18963695, -0.15372306],
       [ 0.90432091, -0.8428519 , -0.5985372 ,  0.5605596 ],
       [ 0.71407786,  0.30963694, -1.04897573, -0.4100024 ],
       [ 0.78197913, -0.70721113, -0.52126199,  0.5632597 ],
       [ 0.13631957, -0.48095938,  0.20854271, -0.79068908],
       [-0.1577882 , -1.13376434,  1.7433439 ,  0.21152392],
       [-1.05176385, -0.4251326 ,  1.78038988, -1.16334579],
       [-0.39045775,  0.76205773, -0.19280482,  0.86299741],
       [ 0.83045195,  1.5261082 , -1.51346033, -1.44319348],
       [ 1.63522659, -0.59442163, -1.87117733, -0.54059556],
       [-0.19547356,  0.8088576 , -0.39945507,  1.00613018],
       [ 1.47338319, -1.91143513, -0.59212223, -1.68341184],
       [ 0.5816877 , -1.28230306,  0.57207388,  1.04378704],
       [-1.7907444 , -0.12770776,  2.23610477, -1.2512715 ],
       [ 0.20538449, -1.11277691,  1.00418807,  0.35848675],
       [ 1.23487961, -1.14568626, -0.92533249, -0.19564073],
       [ 0.28909741,  1.12279225, -0.96593965, -1.65113975],
       [-0.87336971,  0.80863152,  0.2217215 ,  1.64453212],
       [-2.51681459,  2.15804154,  0.62258451,  1.11759127],
       [-0.87231612,  0.37251651,  0.57184616, -0.04615711],
       [-2.92111646,  1.0026191 ,  1.87353293,  1.09221555],
       [-1.13373913,  1.60886683, -0.04982946,  0.42626398],
       [-0.54394322,  1.68906681, -0.52782716,  1.11991839],
       [ 0.47073879, -0.69666227, -0.05041572,  1.07157455],
       [ 1.61774807, -1.51525469, -1.46737816, -0.21171416],
       [-0.64941977, -0.61007121,  1.52214853,  0.36198477],
       [ 0.6664542 , -1.32206668,  0.46481208,  0.49446646],
       [ 0.53864006, -1.17545797,  0.47191021,  1.19890782],
       [ 1.27236054, -0.30838382, -1.45537674,  0.07437403],
       [ 1.18400084, -2.11932978,  1.15175179, -1.40589027],
       [-2.30985569,  0.39717095,  2.03086446,  0.61180711],
       [-0.04001457, -1.17110167,  1.60167673, -0.85038484],
       [-1.73845823,  0.66444358,  1.15294369,  1.14725446],
       [ 1.48955653, -0.32922236, -1.7346557 ,  0.17013387],
       [-1.36555952,  0.59543211,  0.86089293, -0.77654591],
       [-0.64632977, -0.40966498,  1.20121587, -2.10062961],
       [ 0.74724224, -1.67669687,  1.07475276, -1.22894463],
       [ 0.67452121,  0.60552328, -1.12584394, -0.86336873],
       [ 1.006448  , -0.60304048, -0.93910239, -0.87249682],
       [-0.91191209,  1.26043471, -0.03300712,  0.0703043 ],
       [-0.45482085,  0.39702757,  0.11942112, -1.01855398],
       [-1.11045016,  0.7604255 ,  0.47649372,  0.97173373],
       [-0.03077604, -0.85427753,  0.96453541, -0.24352647],
       [ 1.7403336 , -1.36843644, -1.83001783, -1.88495469],
       [-0.04061999,  0.47483987, -0.35599412, -1.3388107 ],
       [ 0.76169366, -0.78206776, -0.42184448,  1.08478719],
       [-2.11057854,  0.48477113,  1.7158041 , -0.19841881],
       [ 0.15778442, -0.87337355,  0.68026595,  1.01755639],
       [ 0.7057278 , -0.17101156, -0.79818137,  0.72117853],
       [-0.43052547, -0.75081822,  1.43385062, -0.39907054],
       [-0.7246411 , -0.68081969,  1.76276324,  1.09906351],
       [ 0.88773086, -1.74494185,  0.86765675, -1.93642471],
       [ 0.56980733,  0.36229851, -0.92074899,  0.34741676],
       [ 0.65052461,  0.78649711, -1.16696468,  1.132557  ],
       [-0.24998484, -0.42288276,  0.67690342,  0.06200291],
       [ 1.07831987,  0.20526777, -1.4016344 ,  0.43057912],
       [-0.63855368, -0.00870915,  0.69079445,  1.27146747],
       [ 1.11322187, -1.61252144, -0.06470889,  0.34574553],
       [ 0.27014859, -0.64981278,  0.20324696,  1.40614044],
       [ 0.33143742, -0.38153102, -0.15820564,  0.13090352],
       [-0.69777465, -0.29641228,  1.11111556, -0.38325154],
       [-0.94836302,  0.76124162,  0.32542697, -1.51658469],
       [-0.26622894,  0.42093207, -0.09176903, -1.81993873],
       [ 1.0565641 , -2.03220383,  1.29269704,  0.60391971],
       [-0.55555626, -0.02230742,  0.60746766,  1.00829761],
       [ 1.5716183 ,  0.07503102, -1.92068017, -0.00935881],
       [ 0.49553738,  0.63858703, -0.96601717, -1.33505216],
       [ 0.03797438, -0.63872352,  0.53924803,  0.26508206],
       [-0.51775296,  1.04663976, -0.24336874,  0.46326337],
       [-0.61478509,  0.71622838,  0.04610925, -1.37466051],
       [ 1.58700538, -1.08457678, -1.63439029,  0.18389275],
       [ 1.11563568,  2.12296507, -1.81337961, -1.71886077],
       [-0.67047576,  0.9285841 , -0.04090955, -0.73395697],
       [-0.69153962, -0.4580254 ,  1.33570254,  2.11506291],
       [ 0.65339445, -1.16740709,  0.23349036, -0.21280828],
       [ 0.39366976, -0.37239382, -0.25000659,  0.99364994],
       [ 0.3254304 , -1.2885287 ,  1.12382922, -1.12607832],
       [ 0.17721286,  0.94839698, -0.80153906,  0.41824272],
       [ 0.67206808, -1.22688985,  0.28966496,  1.49357443],
       [-0.54579093,  0.5828209 ,  0.07302887, -0.45033157],
       [ 1.26206842, -0.65392093, -1.29235702,  0.17093578],
       [ 0.6187755 , -0.72015319, -0.25566572,  1.26764915],
       [ 0.6868576 ,  0.45698456, -1.0813316 , -0.21693104],
       [ 1.70844296, -2.30390445, -0.73242792,  0.45127879],
       [ 0.4655888 , -1.21246444,  0.68223793,  0.37522749],
       [-1.2987197 , -0.30708245,  1.91304508, -1.40542825],
       [ 0.80235108,  0.50329366, -1.21477172,  0.18848095],
       [-0.93719815,  0.50790913,  0.5179197 ,  1.08463468],
       [ 0.48867334, -0.49942125, -0.26996374,  1.44633003],
       [-0.8882693 ,  0.42307713,  0.54273169,  1.68140682],
       [-1.0891347 ,  0.25217792,  0.91952402, -0.60674334],
       [ 0.58920433,  0.80629897, -1.1174037 ,  0.39950719],
       [ 0.84772604, -0.91519952, -0.43550294,  0.53953981],
       [ 0.22463996,  2.3134235 , -1.25462303,  0.76939158],
       [-1.88945126,  1.91938812,  0.33064519,  1.82973958],
       [-1.1554949 ,  1.31659235,  0.1315679 ,  0.22426882],
       [-1.29810642,  0.68364438,  0.71633321,  0.87203084],
       [-1.40129496, -0.04688466,  1.64062685,  0.0247363 ],
       [-1.88708463,  1.14628869,  0.85301957,  0.28613716],
       [-0.73387963,  0.43991229,  0.37077752,  1.91129489],
       [-0.16245857, -0.14238132,  0.25469755,  0.47880723],
       [-0.17348192,  0.86052109, -0.44700531,  1.4026801 ],
       [ 0.75541932,  0.29317675, -1.08592964,  0.87882977],
       [-1.09264141,  0.87580673,  0.37214385, -0.66444457],
       [-0.44986742,  0.57307714, -0.01383488, -0.23548899],
       [ 0.18461941, -0.35467638,  0.01397625,  1.16493204],
       [-1.25138696,  1.5257994 ,  0.08485586,  0.30123222],
       [-0.23566708, -0.24368453,  0.45010222, -0.15483934]])

pca_ = PCA()
pca_.fit(physical_features_scaled_)
plt.title("explained variance by components")
plt.plot(range(1,5), pca_.explained_variance_ratio_.cumsum(), marker='o', linestyle='--')
plt.show()

pca_ = PCA(n_components=2)
pca_.fit_transform(physical_features_scaled_)
loadings_ = pd.DataFrame(data=pca_.components_,
                       columns=physical_features_1.columns,
                       index=['component1','component2'])
fig = plt.figure(figsize=(12,8))
sns.heatmap(loadings_, cmap='RdBu', annot=True, vmin=-1, vmax=1)
plt.title("correlation between components and original physical features, particles over 1 mm^3 ", size=14)
plt.yticks(rotation=45, size=12)
plt.xticks(size=12)
plt.tight_layout()
plt.show()

scores_pca_ = pca_.fit_transform(physical_features_scaled_)
wcss_clust_pca_ = []
for i in range(1,10):
    kmeans = KMeans(i, init="k-means++", random_state=42)
    kmeans.fit(scores_pca_)
    wcss_clust_pca_.append(kmeans.inertia_)
plt.plot(range(1,10), wcss_clust_pca_, marker = '*')
plt.title("pca with k means clustering")
plt.show()

again choosing 3 clusters

k_means_physical_ = KMeans(n_clusters=3, init="k-means++", random_state=42)
k_means_physical_.fit(scores_pca_)

KMeans(n_clusters=3, random_state=42)

k_means_physical_.labels_

array([2, 0, 0, 0, 1, 2, 0, 0, 0, 1, 2, 1, 0, 1, 2, 1, 1, 1, 2, 0, 0, 2,
       1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 2, 0, 0, 2, 0, 1, 1, 1, 0, 0, 2, 2,
       2, 2, 2, 2, 1, 0, 1, 1, 1, 0, 1, 2, 1, 2, 0, 2, 1, 1, 0, 0, 2, 1,
       2, 1, 0, 0, 0, 2, 1, 0, 1, 1, 1, 0, 0, 1, 0, 2, 1, 1, 1, 1, 2, 1,
       1, 2, 0, 0, 1, 2, 2, 0, 0, 2, 2, 1, 1, 1, 0, 1, 2, 0, 1, 0, 0, 1,
       1, 0, 2, 1, 2, 2, 0, 0, 2, 2, 2, 2, 2, 2, 2, 1, 2, 0, 2, 2, 1, 2,
       1])

scores_pca_

array([[ 1.75690042e+00,  4.72780086e-01],
       [-2.56754719e+00,  1.10348903e+00],
       [-1.98661812e+00, -8.85923456e-01],
       [-1.00607094e+00,  1.67612811e+00],
       [ 5.48219592e-01, -4.86792538e-01],
       [ 5.27123498e-01,  1.02123594e+00],
       [-2.30900054e+00,  7.61878451e-01],
       [-1.07213201e+00,  7.36029944e-01],
       [-1.41873762e+00,  5.17691318e-01],
       [-2.13634952e+00, -2.06359115e+00],
       [ 1.23995645e+00,  1.45627575e-01],
       [ 2.26718214e-01, -1.55041316e-01],
       [-2.00167386e+00,  3.25359296e-01],
       [ 1.01969537e+00, -1.22292875e+00],
       [ 1.10335498e+00,  1.99265036e+00],
       [-1.36290135e-01,  1.76631010e-01],
       [-1.80399562e-01, -7.86449910e-01],
       [ 6.45797605e-01, -4.98035960e-01],
       [ 1.11534474e+00,  1.04736782e+00],
       [-1.49688684e+00,  1.75594065e+00],
       [-1.16901911e+00, -2.00639613e-01],
       [ 1.86224844e+00,  1.46862294e+00],
       [ 2.52886181e-01, -1.24562933e+00],
       [-1.66220156e+00, -1.80745821e-01],
       [-5.18705279e-01,  2.19914292e+00],
       [ 3.21524008e-01, -4.44156078e-02],
       [-1.08029136e+00, -1.82428578e-01],
       [-1.07630868e+00,  8.13637313e-01],
       [-9.04199653e-01, -1.25551492e-01],
       [-4.00813036e-01, -6.33176839e-01],
       [ 6.95349159e-01, -1.91458443e+00],
       [ 1.17381987e+00, -1.62332123e+00],
       [ 7.10790218e-01,  8.52531127e-01],
       [-1.29407319e+00,  1.85359457e+00],
       [-2.52143961e+00,  6.05924418e-01],
       [ 5.24426992e-01,  1.03968551e+00],
       [-2.55073025e+00, -1.38990163e+00],
       [-2.40796730e-01, -1.16224251e+00],
       [ 2.01555890e+00, -1.68865482e+00],
       [ 1.01504749e-01, -1.41587851e+00],
       [-1.82723005e+00, -3.41783712e-01],
       [-8.28217611e-01,  1.17419873e+00],
       [ 1.52953495e+00,  7.67532083e-01],
       [ 3.21905669e+00,  1.47248961e+00],
       [ 1.03682217e+00, -7.20964159e-02],
       [ 3.74931497e+00, -1.89470476e-01],
       [ 1.47674393e+00,  1.34176382e+00],
       [ 1.04939267e+00,  1.81382137e+00],
       [-2.74754990e-01, -3.21463606e-01],
       [-2.51930989e+00, -2.91907297e-01],
       [ 1.15876074e+00, -1.34676892e+00],
       [-5.41184495e-01, -1.22024187e+00],
       [-1.77378340e-01, -9.91376704e-01],
       [-1.75430886e+00,  6.73147695e-01],
       [-1.41513136e+00, -2.58620872e+00],
       [ 3.03663126e+00, -8.37272821e-01],
       [ 1.95856631e-01, -2.03708973e+00],
       [ 2.42887504e+00,  1.17006176e-04],
       [-2.03312673e+00,  8.45729803e-01],
       [ 1.39167237e+00, -2.02325309e-01],
       [ 2.97113290e-01, -1.41412672e+00],
       [-9.35138009e-01, -2.16618010e+00],
       [-1.12457939e+00,  1.01022752e+00],
       [-1.69042842e+00, -3.26992756e-02],
       [ 1.09542986e+00,  1.00264076e+00],
       [ 2.10596202e-01,  5.88090966e-02],
       [ 1.60688130e+00,  4.59276096e-01],
       [ 1.53189468e-01, -1.29448510e+00],
       [-3.26258105e+00, -2.40521436e-01],
       [-4.05043349e-01,  3.56059367e-01],
       [-7.02662564e-01, -1.55437131e-01],
       [ 2.50987195e+00, -7.14153871e-01],
       [ 2.52863195e-01, -9.18258393e-01],
       [-7.58108167e-01,  4.83810833e-01],
       [ 6.72167922e-01, -1.53105345e+00],
       [ 1.54165404e+00, -1.42428204e+00],
       [-1.38651977e+00, -2.21115463e+00],
       [-6.53805386e-01,  9.04493428e-01],
       [-4.48516121e-01,  1.51483573e+00],
       [ 4.03768449e-01, -7.31779455e-01],
       [-1.29728551e+00,  1.09401610e+00],
       [ 1.20651745e+00, -2.14370014e-01],
       [-1.28513436e+00, -1.14111036e+00],
       [ 1.20973118e-01, -3.84931597e-01],
       [-4.12371376e-01, -1.73123540e-01],
       [ 8.56918323e-01, -9.77888554e-01],
       [ 6.43558352e-01,  1.27807844e-01],
       [-2.73626883e-01,  7.01654120e-02],
       [-5.98636052e-01, -2.26338064e+00],
       [ 1.01764572e+00, -2.18483950e-01],
       [-2.10229819e+00,  1.23863135e+00],
       [-1.04762324e+00,  8.56948719e-01],
       [ 1.13744774e-01, -7.78626406e-01],
       [ 7.51206145e-01,  1.03409634e+00],
       [ 2.87900466e-01,  2.88730392e-01],
       [-2.31137617e+00,  2.08549714e-01],
       [-1.52916347e+00,  2.44604331e+00],
       [ 5.54468691e-01,  6.14870928e-01],
       [ 1.68594790e+00, -8.14836537e-01],
       [-8.17777040e-01, -1.07859220e+00],
       [-2.34952618e-01,  3.77864387e-02],
       [-4.41485323e-01, -1.87913118e+00],
       [-8.41398991e-02,  1.29297676e+00],
       [-2.94997024e-01, -8.67894829e-01],
       [ 4.91813592e-01,  3.27316965e-01],
       [-1.75310932e+00,  3.24522391e-01],
       [-4.35474390e-01, -1.79293760e-01],
       [-9.62433942e-01,  9.79411953e-01],
       [-2.26980987e+00, -1.23954593e+00],
       [-2.82487949e-01, -1.29071115e+00],
       [ 1.38561125e+00, -1.65498532e+00],
       [-9.72976746e-01,  1.16606142e+00],
       [ 1.45169155e+00,  2.58455891e-01],
       [-2.17696000e-01,  2.97190328e-02],
       [ 1.58460194e+00,  2.79680395e-01],
       [ 1.15795613e+00, -4.74362863e-01],
       [-5.98528084e-01,  1.37447758e+00],
       [-9.86324500e-01, -3.42351892e-01],
       [ 2.33235176e-01,  2.67608960e+00],
       [ 2.75634554e+00,  1.58810714e+00],
       [ 1.42223378e+00,  9.71437725e-01],
       [ 1.80837260e+00,  2.35894952e-01],
       [ 1.84874157e+00, -1.04147738e+00],
       [ 2.28008616e+00,  4.09129549e-01],
       [ 1.46155719e+00,  4.37541073e-01],
       [ 3.47131366e-01, -1.84167841e-01],
       [ 6.24571116e-01,  1.17622820e+00],
       [-7.32524173e-01,  1.04374637e+00],
       [ 1.07425808e+00,  3.32892113e-01],
       [ 4.41015652e-01,  4.09822071e-01],
       [ 1.11475984e-01, -8.21270618e-02],
       [ 1.56318184e+00,  1.17418846e+00],
       [ 2.70363519e-01, -4.90783873e-01]])

df_pca_scores_ = pd.DataFrame(scores_pca_, columns=['Component 1', 'Component 2'])

k_means_pca_clusters_ = pd.DataFrame(k_means_physical_.labels_, columns=["PcaClusters"])
df_segm_pca_ = pd.concat([particles_over_1.reset_index(drop=True), df_pca_scores_ , k_means_pca_clusters_], axis=1)

df_segm_pca_

	Volume3d (mm^3)	BaryCenterX (mm)	BaryCenterY (mm)	BaryCenterZ (mm)	Anisotropy	Elongation	Flatness	VoxelFaceArea (mm^2)	Sphericity	SurfaceParticles	Component 1	Component 2	PcaClusters
0	91.75290	-1249.26	-1243.58	-1685.28	0.545994	0.731435	0.620706	244.73600	0.401975	False	1.756900	0.472780	2
1	33.28520	-1241.94	-1257.46	-1686.12	0.946436	0.566783	0.094506	175.07800	0.285816	False	-2.567547	1.103489	0
2	26.85760	-1246.98	-1235.86	-1648.17	0.862477	0.343091	0.400835	194.94600	0.222475	False	-1.986618	-0.885923	0
3	20.57550	-1249.24	-1250.28	-1680.69	0.844642	0.736602	0.210912	94.27890	0.385156	False	-1.006071	1.676128	0
4	17.93760	-1242.87	-1249.47	-1647.25	0.710594	0.471159	0.614242	71.30980	0.464705	False	0.548220	-0.486793	1
...	...	...	...	...	...	...	...	...	...	...	...	...	...
128	1.03507	-1238.85	-1242.61	-1697.13	0.606199	0.684806	0.575054	13.58990	0.364123	False	1.074258	0.332892	2
129	1.03052	-1249.60	-1238.66	-1687.48	0.687950	0.629907	0.495391	12.33400	0.400023	False	0.441016	0.409822	2
130	1.00715	-1240.22	-1230.60	-1689.03	0.768647	0.461662	0.501131	9.39435	0.517226	False	0.111476	-0.082127	1
131	1.00496	-1241.06	-1238.86	-1681.39	0.586009	0.802680	0.515760	10.90470	0.444942	False	1.563182	1.174188	2
132	1.00493	-1237.08	-1242.46	-1682.74	0.715193	0.481790	0.591144	11.92770	0.406772	False	0.270364	-0.490784	1

133 rows × 13 columns

df_mean_segm_pca_ = df_segm_pca_.groupby('PcaClusters').mean()

df_mean_segm_pca_

	Volume3d (mm^3)	BaryCenterX (mm)	BaryCenterY (mm)	BaryCenterZ (mm)	Anisotropy	Elongation	Flatness	VoxelFaceArea (mm^2)	Sphericity	SurfaceParticles	Component 1	Component 2
PcaClusters
0	5.269345	-1246.889302	-1249.489535	-1669.974419	0.868871	0.516571	0.262788	37.770130	0.388150	0.0	-1.425376	0.598309
1	3.278162	-1245.903333	-1248.813750	-1675.851667	0.752138	0.391549	0.639643	25.457580	0.414491	0.0	0.041122	-1.003934
2	4.874370	-1245.501429	-1245.539048	-1673.814524	0.610548	0.689252	0.577715	25.373691	0.458051	0.0	1.412316	0.534800

fig = plt.figure(figsize=(12,8))

plt.subplot(2, 2, 1)
sns.boxplot(data=df_segm_pca_, x='PcaClusters',  y='Anisotropy')
plt.title("Anisotropy")
plt.xlabel('clusters', {'size':12})
plt.ylabel('Anisotropy', {'size':12})

plt.subplot(2, 2, 2)
sns.boxplot(data=df_segm_pca_, x='PcaClusters',  y='Elongation')
plt.title("Elongation")
plt.xlabel('clusters', {'size':12})
plt.ylabel('Elongation', {'size':12})

plt.subplot(2, 2, 3)
sns.boxplot(data=df_segm_pca_, x='PcaClusters',  y='Flatness')
plt.title("Flatness")
plt.xlabel('clusters', {'size':12})
plt.ylabel('Flatness', {'size':12})

plt.subplot(2, 2, 4)
sns.boxplot(data=df_segm_pca_, x='PcaClusters',  y='Sphericity')
plt.title("Sphericity")
plt.xlabel('clusters', {'size':12})
plt.ylabel('Sphericity', {'size':12})
plt.tight_layout(pad=2)
plt.show()

import plotly.express as px

fig = px.scatter_3d(df_segm_pca_, x='BaryCenterX (mm) ', y='BaryCenterY (mm) ', z='BaryCenterZ (mm) ',
              color='PcaClusters', symbol='PcaClusters', opacity=0.6)
fig.update_layout(margin=dict(l=0, r=0, b=0, t=0))
fig.show()

as we can see above if considering particles greater than or equal to 1 mm^3, no significant cluster appear

3. Slicing venus based on vertical coordinates and exploring each slice

total height of the figurine is (cca 114.099 mm), figure was dived into 6 even parts (cca 19.016 mm) along vertical axis, using BarycenterZ coordinates

import plotly.express as px

fig = px.scatter_3d(all_particles, x='BaryCenterX (mm) ', y='BaryCenterY (mm) ', z='BaryCenterZ (mm) ', opacity=0.6)
fig.update_layout(margin=dict(l=0, r=0, b=0, t=0))
fig.show()

very_bottom_part = all_particles['BaryCenterZ (mm) '].min()
very_upper_part = all_particles['BaryCenterZ (mm) '].max()

range_z = abs(very_bottom_part) - abs(very_upper_part)
part = range_z/6
print(f"height {range_z}")
print(f"one part height {part}")

height 114.09999999999991
one part height 19.01666666666665

# creating all parts
first = (very_bottom_part, very_bottom_part + part)
second = (first[1], first[1] + part)
third = (second[1], second[1] + part)
fourth = (third[1], third[1] + part)
fifth = (fourth[1], fourth[1] + part)
sixth = (fifth[1], fifth[1] + part)

all_parts = [first, second, third, fourth, fifth, sixth]

intervals for all designated slices

all_parts[::-1]

[(-1636.6866666666667, -1617.67),
 (-1655.7033333333334, -1636.6866666666667),
 (-1674.72, -1655.7033333333334),
 (-1693.7366666666667, -1674.72),
 (-1712.7533333333333, -1693.7366666666667),
 (-1731.77, -1712.7533333333333)]

all_particles_parts = []
for part in all_parts[::-1]:
    all_particles_parts.append(all_particles[(all_particles['BaryCenterZ (mm) '] >= part[0]) & (all_particles['BaryCenterZ (mm) '] <= part[1])])

very top (head) - part 1

part_1 = all_particles_parts[0]
part_1.describe()

	Volume3d (mm^3)	BaryCenterX (mm)	BaryCenterY (mm)	BaryCenterZ (mm)	Anisotropy	Elongation	Flatness	VoxelFaceArea (mm^2)	Sphericity
count	10262.000000	10262.000000	10262.000000	10262.000000	10262.000000	10262.000000	10262.000000	10262.000000	10262.000000
mean	0.005425	-1246.557975	-1248.402321	-1628.567112	0.694496	0.412995	0.363506	0.118019	0.660916
std	0.095298	3.464295	3.071000	4.889973	0.386088	0.378781	0.514893	0.822170	0.079196
min	0.000007	-1254.640000	-1255.920000	-1636.680000	-1.000000	-1.000000	-1.000000	0.002151	0.211992
25%	0.000048	-1249.150000	-1250.690000	-1632.680000	0.623181	0.249931	0.216478	0.009320	0.617429
50%	0.000183	-1246.795000	-1248.470000	-1628.870000	0.783372	0.451832	0.491374	0.024374	0.660833
75%	0.000672	-1243.850000	-1246.122500	-1624.740000	0.904537	0.657428	0.699558	0.059502	0.707021
max	7.347280	-1237.990000	-1240.860000	-1617.670000	1.000000	0.999919	1.000000	46.027800	0.805997

count_over_1 = part_1[part_1['Volume3d (mm^3) '] > 2].count()[0]
res_1 = round(count_over_1/part_1.count()[0]*100,3)
print(f"in first part only {res_1} % particles are larger than 0.1 mm^3 ")

in first part only 0.039 % particles are larger than 0.1 mm^3 

part 2 upper torso

part_2 = all_particles_parts[1]
part_2.describe()

	Volume3d (mm^3)	BaryCenterX (mm)	BaryCenterY (mm)	BaryCenterZ (mm)	Anisotropy	Elongation	Flatness	VoxelFaceArea (mm^2)	Sphericity
count	17142.000000	17142.000000	17142.000000	17142.000000	17142.000000	17142.000000	17142.000000	17142.000000	17142.000000
mean	0.011573	-1246.419391	-1248.208249	-1647.486087	0.697093	0.428800	0.350486	0.154942	0.662480
std	0.331032	3.076190	6.975334	5.213213	0.377345	0.373879	0.510058	2.211765	0.080122
min	0.000007	-1256.190000	-1265.670000	-1655.700000	-1.000000	-1.000000	-1.000000	0.002151	0.210321
25%	0.000041	-1248.660000	-1253.140000	-1651.990000	0.620390	0.260574	0.201926	0.008603	0.620853
50%	0.000149	-1246.380000	-1248.230000	-1647.930000	0.776863	0.469232	0.471563	0.020790	0.664293
75%	0.000529	-1244.000000	-1243.180000	-1643.450000	0.907883	0.667070	0.680712	0.049466	0.707043
max	26.857600	-1238.690000	-1232.030000	-1636.690000	1.000000	0.999919	1.000000	194.946000	0.805997

count_over_2 = part_2[part_2['Volume3d (mm^3) '] > 2].count()[0]
res_2 = round(count_over_2/part_2.count()[0]*100,3)
print(f"in second part only {res_2} % particles are larger than 0.1 mm^3 ")

in second part only 0.082 % particles are larger than 0.1 mm^3 

part 3 - belly and breasts

part_3 = all_particles_parts[2]
part_3.describe()

	Volume3d (mm^3)	BaryCenterX (mm)	BaryCenterY (mm)	BaryCenterZ (mm)	Anisotropy	Elongation	Flatness	VoxelFaceArea (mm^2)	Sphericity
count	27812.000000	27812.000000	27812.000000	27812.000000	27812.000000	27812.000000	27812.000000	27812.000000	27812.000000
mean	0.007908	-1247.887187	-1247.471837	-1665.837904	0.682229	0.436214	0.335999	0.137669	0.658013
std	0.166526	5.373476	8.251930	5.465757	0.402690	0.393944	0.509167	1.166347	0.080487
min	0.000007	-1262.260000	-1267.530000	-1674.720000	-1.000000	-1.000000	-1.000000	0.002151	0.243926
25%	0.000048	-1251.900000	-1254.440000	-1670.600000	0.610563	0.285205	0.185887	0.008603	0.613080
50%	0.000163	-1247.680000	-1246.710000	-1666.175000	0.773096	0.499411	0.453831	0.021507	0.660065
75%	0.000604	-1243.820000	-1241.000000	-1661.250000	0.904845	0.680393	0.666915	0.055201	0.707021
max	16.360700	-1234.030000	-1227.770000	-1655.710000	1.000000	0.999919	1.000000	83.427600	0.805997

count_over_3 = part_3[part_3['Volume3d (mm^3) '] > 2].count()[0]
res_3 = round(count_over_3/part_3.count()[0]*100,3)
print(f"in second part only {res_3} % particles are larger than 0.1 mm^3 ")

in second part only 0.072 % particles are larger than 0.1 mm^3 

part 4 - bottom part of belly and bottom

part_4 = all_particles_parts[3]
part_4.describe()

	Volume3d (mm^3)	BaryCenterX (mm)	BaryCenterY (mm)	BaryCenterZ (mm)	Anisotropy	Elongation	Flatness	VoxelFaceArea (mm^2)	Sphericity
count	23683.000000	23683.000000	23683.000000	23683.000000	23683.000000	23683.000000	23683.000000	23683.000000	23683.000000
mean	0.014307	-1245.832862	-1244.941928	-1682.686778	0.660273	0.435814	0.293013	0.173844	0.655450
std	0.657165	5.621560	10.259573	5.448225	0.455162	0.436807	0.515325	2.266253	0.081953
min	0.000007	-1259.100000	-1269.460000	-1693.730000	-1.000000	-1.000000	-1.000000	0.002151	0.174676
25%	0.000048	-1250.510000	-1253.230000	-1687.190000	0.607987	0.316045	0.140945	0.009319	0.610433
50%	0.000163	-1245.860000	-1243.900000	-1681.580000	0.776442	0.513000	0.411881	0.022224	0.658454
75%	0.000628	-1241.200000	-1236.020000	-1678.130000	0.911574	0.702300	0.627665	0.057352	0.707018
max	91.752900	-1233.450000	-1225.980000	-1674.720000	1.000000	0.999980	1.000000	244.736000	0.805997

count_over_4 = part_4[part_4['Volume3d (mm^3) '] > 2].count()[0]
res_4 = round(count_over_4/part_4.count()[0]*100,3)
print(f"in second part only {res_4} % particles are larger than 0.1 mm^3 ")

in second part only 0.063 % particles are larger than 0.1 mm^3 

part 5: upper part of legs

part_5 = all_particles_parts[4]
part_5.describe()

	Volume3d (mm^3)	BaryCenterX (mm)	BaryCenterY (mm)	BaryCenterZ (mm)	Anisotropy	Elongation	Flatness	VoxelFaceArea (mm^2)	Sphericity
count	16587.000000	16587.000000	16587.000000	16587.000000	16587.000000	16587.000000	16587.000000	16587.000000	16587.000000
mean	0.007997	-1245.845139	-1246.627532	-1702.859876	0.663054	0.459394	0.319604	0.136685	0.656183
std	0.145740	3.962872	6.345024	5.457670	0.432962	0.424656	0.494787	1.029505	0.077958
min	0.000007	-1255.450000	-1263.140000	-1712.750000	-1.000000	-1.000000	-1.000000	0.002151	0.271719
25%	0.000054	-1249.130000	-1251.820000	-1707.530000	0.602599	0.333270	0.171964	0.010037	0.612787
50%	0.000183	-1245.890000	-1246.470000	-1702.710000	0.764311	0.532567	0.438477	0.023657	0.656347
75%	0.000645	-1242.570000	-1241.130000	-1698.080000	0.899015	0.723497	0.633601	0.058068	0.704103
max	8.926760	-1236.750000	-1232.320000	-1693.740000	1.000000	0.999955	1.000000	47.584900	0.805997

count_over_5 = part_5[part_5['Volume3d (mm^3) '] > 2].count()[0]
res_5 = round(count_over_5/part_5.count()[0]*100,3)
print(f"in second part only {res_5} % particles are larger than 0.1 mm^3 ")

in second part only 0.072 % particles are larger than 0.1 mm^3 

lower part of legs, part 6

part_6 = all_particles_parts[5]
part_6.describe()

	Volume3d (mm^3)	BaryCenterX (mm)	BaryCenterY (mm)	BaryCenterZ (mm)	Anisotropy	Elongation	Flatness	VoxelFaceArea (mm^2)	Sphericity
count	7046.000000	7046.000000	7046.000000	7046.000000	7046.000000	7046.000000	7046.000000	7046.000000	7046.000000
mean	0.007143	-1245.257610	-1245.954489	-1718.462267	0.660241	0.469356	0.330794	0.139183	0.656312
std	0.104226	3.246921	4.492561	3.860653	0.421707	0.418355	0.489197	1.020935	0.078268
min	0.000007	-1253.260000	-1256.650000	-1731.770000	-1.000000	-1.000000	-1.000000	0.002151	0.167262
25%	0.000061	-1247.827500	-1250.170000	-1721.330000	0.602826	0.341241	0.216774	0.010753	0.614265
50%	0.000204	-1245.300000	-1244.570000	-1717.925000	0.753106	0.542593	0.449330	0.025808	0.657366
75%	0.000692	-1242.600000	-1242.370000	-1715.190000	0.885106	0.728761	0.632880	0.060220	0.703510
max	4.631700	-1237.830000	-1237.340000	-1712.760000	1.000000	0.999919	1.000000	50.540500	0.805997

count_over_6 = part_6[part_6['Volume3d (mm^3) '] > 2].count()[0]
res_6 = round(count_over_6/part_6.count()[0]*100,3)
print(f"in second part only {res_6} % particles are larger than 0.1 mm^3 ")

in second part only 0.057 % particles are larger than 0.1 mm^3 

venus_particle_distrubution_towards_vertical_axis = pd.DataFrame([res_1, res_2,res_3,res_4,res_5,res_6], 
                                                                 columns=["particles over 2 mm^3 [%]"],
                                                                index=['part 1 (head)','part 2','part 3','part 4','part 5','part 6 (legs)'])

table = ratio of particles with the volume greater than or equal to 2 mm^3 to all particles in a particular part of the venus

venus_particle_distrubution_towards_vertical_axis.values

array([[0.039],
       [0.082],
       [0.072],
       [0.063],
       [0.072],
       [0.057]])

As we can see above even dividing venus into 6 even vertical parts using BaryCenterZ coordinates, does not bring any significant results regarding distribution larger particles greater than or equal to 2 mm^3

part_1['slice'] = 1
part_2['slice'] = 2
part_3['slice'] = 3
part_4['slice'] = 4
part_5['slice'] = 5
part_6['slice'] = 6

C:\Users\hamro\Anaconda3\lib\site-packages\pandas\core\frame.py:3607: SettingWithCopyWarning:


A value is trying to be set on a copy of a slice from a DataFrame.
Try using .loc[row_indexer,col_indexer] = value instead

See the caveats in the documentation: https://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy

mean values of physical features and volume fo each slice seperately (for all particles)

all_slices = pd.concat([part_1, part_2, part_3, part_4, part_5, part_6], axis=0)
slices_means = all_slices.groupby('slice').mean()[['Volume3d (mm^3) ', 'Anisotropy', 'Elongation', 'Flatness', 'Sphericity']]

slices_means

	Volume3d (mm^3)	Anisotropy	Elongation	Flatness	Sphericity
slice
1	0.005425	0.694496	0.412995	0.363506	0.660916
2	0.011573	0.697093	0.428800	0.350486	0.662480
3	0.007908	0.682229	0.436214	0.335999	0.658013
4	0.014307	0.660273	0.435814	0.293013	0.655450
5	0.007997	0.663054	0.459394	0.319604	0.656183
6	0.007143	0.660241	0.469356	0.330794	0.656312

plt.figure(figsize=(12,8))
plt.title("physical features for each slice", {'size':15})
sns.heatmap(slices_means[['Anisotropy', 'Elongation', 'Flatness', 'Sphericity']], annot=True)
plt.ylabel("slize", {'size':14})
plt.xlabel("features", {'size':14})
plt.show()

plt.figure(figsize=(10,8))
plt.bar(x=slices_means.index, height=slices_means['Volume3d (mm^3) '])
plt.title("total volume of particles in each slice", {'size':14})
plt.xlabel("slices", {'size':12})
plt.ylabel("mean volume [mm^3]", {'size':12})
plt.show()

mean values of physical features and volume fo each slice seperately (only for particles with volume greater than or equal to 2 mm^3)

all_slices_part_over_2 = pd.concat([part_1[part_1['Volume3d (mm^3) '] > 2], part_2[part_2['Volume3d (mm^3) '] > 2], part_3[part_3['Volume3d (mm^3) '] > 2], part_4[part_4['Volume3d (mm^3) '] > 2], part_5[part_5['Volume3d (mm^3) '] > 2], part_6[part_6['Volume3d (mm^3) '] > 2]], axis=0)

all_slices_part_over_2_means = all_slices_part_over_2.groupby('slice').mean()[['Volume3d (mm^3) ', 'Anisotropy', 'Elongation', 'Flatness', 'Sphericity']]

count_for_each_slice = all_slices_part_over_2.groupby('slice').count()

final_interval = []
for i in all_parts[::-1]:
    final_interval.append((round(i[1],2), round(i[0],2)))
final_interval

[(-1617.67, -1636.69),
 (-1636.69, -1655.7),
 (-1655.7, -1674.72),
 (-1674.72, -1693.74),
 (-1693.74, -1712.75),
 (-1712.75, -1731.77)]

 venus_particle_distrubution_towards_vertical_axis['particles over 2 mm^3 [%]'].values

array([0.039, 0.082, 0.072, 0.063, 0.072, 0.057])

all_slices_part_over_2_means['No of particles'] = count_for_each_slice['Volume3d (mm^3) ']
all_slices_part_over_2_means['BarycenterZ(range)'] = final_interval
all_slices_part_over_2_means['ration particles over 2 mm^3 to all particles [%]'] = venus_particle_distrubution_towards_vertical_axis['particles over 2 mm^3 [%]'].values
all_slices_part_over_2_means

	Volume3d (mm^3)	Anisotropy	Elongation	Flatness	Sphericity	No of particles	BarycenterZ(range)	ration particles over 2 mm^3 to all particles [%]
slice
1	3.712422	0.701612	0.569758	0.512672	0.388485	4	(-1617.67, -1636.69)	0.039
2	9.042826	0.783831	0.497701	0.460212	0.366160	14	(-1636.69, -1655.7)	0.082
3	4.582709	0.767979	0.539119	0.451089	0.406029	20	(-1655.7, -1674.72)	0.072
4	12.840997	0.750144	0.512394	0.474677	0.394590	15	(-1674.72, -1693.74)	0.063
5	4.613992	0.724846	0.515641	0.552434	0.439833	12	(-1693.74, -1712.75)	0.072
6	3.832298	0.657946	0.534601	0.646867	0.395606	4	(-1712.75, -1731.77)	0.057

plt.figure(figsize=(12,8))
plt.title("physical features for each slice, only particles with volume over 2 mm^3", {'size':15})
sns.heatmap(all_slices_part_over_2_means[['Anisotropy', 'Elongation', 'Flatness', 'Sphericity']], annot=True)
plt.ylabel("slize", {'size':14})
plt.xlabel("features", {'size':14})
plt.show()

plt.figure(figsize=(10,8))
plt.bar(x=all_slices_part_over_2_means.index, height=all_slices_part_over_2_means['Volume3d (mm^3) '])
plt.title("total volume of particles (over 2 mm^3) in each slice", {'size':14})
plt.xlabel("slices", {'size':12})
plt.ylabel("mean volume [mm^3]", {'size':12})
plt.show()

venus slices highlighed on 3d plot

import plotly.express as px

fig = px.scatter_3d(all_slices, x='BaryCenterX (mm) ', y='BaryCenterY (mm) ', z='BaryCenterZ (mm) ',
              color='slice', opacity=0.6)
fig.update_layout(margin=dict(l=0, r=0, b=0, t=0))
fig.show()